Twisted conjugacy classes for polyfree groups

TL;DR

This paper investigates the properties of automorphisms and Reidemeister classes in finitely generated polyfree groups, revealing conditions under which automorphisms have infinite Reidemeister numbers, especially when the group has nonzero Euler characteristic.

Contribution

It establishes new results on the Reidemeister numbers of automorphisms in polyfree groups, including conditions for infiniteness and structural properties of automorphism groups.

Findings

Automorphisms in certain polyfree groups have infinite Reidemeister numbers.

Groups with nonzero Euler characteristic have automorphism subgroups with infinite Reidemeister numbers.

For some length-2 polyfree groups, all automorphisms have infinitely many Reidemeister classes.

Abstract

Let $G$ be a finitely generated polyfree group. If $G$ has nonzero Euler characteristic then we show that $Aut(G)$ has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain $G$ of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.